The theory of compressive sensing, also known as compressed sensing or CS is a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals from far fewer samples or measurements than traditional methods that follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (the so-called Nyquist rate). Alternatively, the sampling rate required by CS is directly proportional to the sparsity level of the signal on a proper basis. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest (usually on a selective basis), and incoherence, which pertains to the sensing modality.
Sparsity expresses the idea that the “information rate” of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom, which is comparably much smaller than its (finite) length. Sparsity says that objects have a sparse representation in one domain can be spread out in the domain in which they are acquired, such as a Dirac or spike in the time domain can be spread out in the frequency domain. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise representations when expressed on an appropriate basis.
Incoherence extends the duality between time and frequency. Incoherence says that each measurement of the signal should have low or limited correlation between all the other measurements regardless of the signal basis such that each measurement can carry enough new information about the signal, or in other words, the measurements carry limited mutual information. Therefore, the redundancy in the measurements is limited, and the sampling rate required is directly proportional to the amount of information desired regardless of the signal basis. Incoherence extends the proportionality between the sampling rate and the signal information to an arbitrary basis, unlike the Nyquist rate, which is proportional to the signal information on Fourier basis only.
According to the theory of CS, one can design efficient sensing or sampling protocols that capture the useful information content embedded in a sparse signal and condense it into a small amount of data. These protocols are nonadaptive and simply involve correlating the signal with randomness that are incoherent with the sparsifying basis. What is most remarkable about these sampling protocols is that they allow a sensor to very efficiently capture the information in a sparse signal without trying to comprehend the signal. Further, there is a way to use numerical optimization to reconstruct the full-length signal from the small amount of collected data. In other words, systems based on CS principles can sample—in a signal independent fashion—at a low rate and later use computation power for reconstruction. Effectively, such systems sense and compress data simultaneously (thus the name compressed sensing).
CS has received significant research attention due to its insights into sampling and signal processing. The CS framework offers an approach for compressive data sampling at sub-Nyquist rates, which may lower power consumption and the costs associated with data acquisition systems for a variety of signals. CS techniques have shown promising results in a wide range of applications, including the data acquisition in bio-medical applications, such as (but not limited to) magnetic resonance imaging (MRI), electrocardiogram (ECG), and electroencephalogram (EEG).
However, CS techniques in real applications still face significant challenges. For example, the digital reconstruction in CS typical involves solving an optimization problem using either greedy heuristics or convex relaxations. The computational complexity of these processes is usually 1-2 orders of magnitude higher than the orthogonal transform used in the Nyquist framework. This can lead to a low efficiency of real-time processing on general purpose processors, limiting the application of CS techniques on mobile platforms. An efficient hardware solution may benefit a variety of CS applications in terms of cost, portability, and battery-life.